We establish sharp Liouville theorems for the integral equation
$ u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n, $
where
$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $
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[1] |
D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
![]() ![]() |
[2] |
P. d'Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447–1476.
doi: 10.1142/S0218202515500384.![]() ![]() ![]() |
[3] |
P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira, Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38-53.
doi: 10.1016/j.na.2017.08.005.![]() ![]() ![]() |
[4] |
J. Bertoin, Lévy Processes, Cambridge University Press, 1996.
![]() ![]() |
[5] |
J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N.![]() ![]() ![]() |
[6] |
L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903.![]() ![]() ![]() |
[7] |
D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.
doi: 10.1017/prm.2018.67.![]() ![]() ![]() |
[8] |
G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3.![]() ![]() ![]() |
[9] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116.![]() ![]() ![]() |
[10] |
W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.
doi: 10.1081/PDE-200044445.![]() ![]() ![]() |
[11] |
W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5.![]() ![]() ![]() |
[12] |
P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of lecture Notes in Math. 1–43, Springer, Berlin, 2006.
doi: 10.1007/11545989_1.![]() ![]() ![]() |
[13] |
W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.
doi: 10.3934/dcds.2018117.![]() ![]() ![]() |
[14] |
P. Le, Symmetry and classification of solutions to an integral equation of Choquard type, submitted for publication.
![]() |
[15] |
P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141.
doi: 10.1016/j.na.2019.03.006.![]() ![]() ![]() |
[16] |
Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.
doi: 10.3934/dcds.2018236.![]() ![]() ![]() |
[17] |
Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.
doi: 10.1007/s00209-012-1036-6.![]() ![]() ![]() |
[18] |
E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032.![]() ![]() ![]() |
[19] |
E. Lieb, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977) 185–194.
![]() ![]() |
[20] |
S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.
doi: 10.1016/j.na.2009.01.014.![]() ![]() ![]() |
[21] |
L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3.![]() ![]() ![]() |
[22] |
I. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019.![]() ![]() ![]() |
[23] |
V. Moroz and J. V. Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.
doi: 10.1007/s11784-016-0373-1.![]() ![]() ![]() |
[24] |
V. Moroz and J. V. Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains}, J. Differential Equations, 254 (2013), 3089-3145.
doi: 10.1016/j.jde.2012.12.019.![]() ![]() ![]() |
[25] |
S. Pekar, Untersuchungen über die Elekronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
![]() |
[26] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970.
![]() ![]() |
[27] |
V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005.![]() ![]() ![]() |
[28] |
D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.
doi: 10.1016/j.aml.2014.12.007.![]() ![]() ![]() |
[29] |
W. Zhang and X. Wu, Nodal solutions for a fractional Choquard equation, J. Math. Anal. Appl., 464 (2018), 1167-1183.
doi: 10.1016/j.jmaa.2018.04.048.![]() ![]() ![]() |